Track first-success chances with geometric results and summaries. Download tables, print reports, and validate assumptions. Built for planners, analysts, students, and probability practice needs.
Example inputs use p = 0.25 and trials 1 through 6.
| Trial | Exact Probability | Cumulative Probability | At Least x Trials |
|---|---|---|---|
| 1 | 0.25 | 0.25 | 1 |
| 2 | 0.1875 | 0.4375 | 0.75 |
| 3 | 0.140625 | 0.578125 | 0.5625 |
| 4 | 0.10546875 | 0.68359375 | 0.421875 |
| 5 | 0.0791015625 | 0.7626953125 | 0.31640625 |
| 6 | 0.0593261719 | 0.8220214844 | 0.2373046875 |
The calculator assumes the first success happens on trial x.
Exact Probability: P(X = x) = (1 - p)x - 1 × p
Cumulative Probability: P(X ≤ x) = 1 - (1 - p)x
At Least x Trials: P(X ≥ x) = (1 - p)x - 1
Mean: 1 ÷ p
Variance: (1 - p) ÷ p2
Standard Deviation: √Variance
Mode: 1
A geometric distribution calculator estimates the chance that the first success occurs on a chosen trial. It works with repeated and independent trials. Each trial must share the same success probability. The tool measures waiting time until the first success. That makes it useful for probability classes, testing workflows, and repeated attempt analysis.
Many real tasks repeat until one event happens. A support team may wait for the first answered call. A marketer may track the first click after repeated views. A tester may repeat checks until the first pass. In each case, the geometric model gives a simple way to describe trial-based uncertainty.
This page returns the exact probability for the first success on trial x. It also shows the cumulative probability up to trial x. You can review the chance that at least x trials are needed. The calculator also reports mean, variance, standard deviation, and mode. A generated table helps compare values across several trials.
A larger success probability moves more weight toward earlier trials. A smaller probability spreads the waiting time over more trials. If the exact probability is highest at small trial values, success is likely early. If the cumulative result rises slowly, the event may take longer than expected. Mean and variance help summarize timing and spread.
Students use this tool to practice discrete probability. Analysts use it to estimate first-response timing. Engineers use it for inspection and reliability studies. Product teams may model first conversion timing. Operations teams may review repeated attempts before one success. The export options help when results need to be shared in reports or worksheets.
The model assumes trial independence and a constant success probability. If those assumptions fail, the output may not match reality. Use realistic values and compare multiple trial numbers. Review the example table before making decisions. That simple step improves interpretation, supports better communication, and makes geometric distribution results easier to trust.
It measures the probability that the first success occurs on a specific trial during repeated independent trials with the same success probability.
Use it when each trial has only two outcomes, trials are independent, and the success probability stays constant from one trial to the next.
The exact output gives the chance that the first success happens exactly on trial x. It uses the geometric probability mass formula.
Cumulative probability shows the chance that the first success happens on or before the selected trial number. It combines outcomes up to x.
For the standard geometric distribution, the highest single-trial probability occurs at the first trial. That makes the mode equal to 1.
It means the first success has not happened before trial x. In other words, you need x or more trials to see the first success.
Yes. The calculator includes a CSV download for the generated table and a print-based PDF option for saving or sharing outputs.
The probability p must be greater than 0 and less than 1. Values outside that range do not fit the geometric model.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.